Normal Probability Plot — Definition, Formula & Examples
A normal probability plot is a graph that plots each data value in your sample against the value it would be expected to have if the data were perfectly normally distributed. If the points fall close to a straight line, the data are approximately normal.
A normal probability plot (also called a normal quantile plot or Q-Q plot) displays the ordered observations of a dataset on one axis against the corresponding theoretical quantiles (z-scores) of a standard normal distribution on the other axis. Approximate normality is indicated when the plotted points closely follow a linear pattern.
How It Works
To construct the plot, first sort your data values from smallest to largest. For each data value, calculate its expected normal score — the z-score that corresponds to its percentile rank in the dataset. Plot each data point against its expected z-score. If the data come from a normal distribution, the points will form a roughly straight line. Curves or systematic bends reveal departures from normality: an S-shape suggests heavy or light tails, while a curve bending upward on the right suggests right skew.
Worked Example
Problem: A sample of 5 quiz scores is: 62, 71, 76, 82, 94. Determine whether the data appear approximately normal using a normal probability plot.
Step 1: Order the data and assign percentile positions: The data sorted are 62, 71, 76, 82, 94. For 5 values, a common rule assigns the -th value the percentile .
Step 2: Find the corresponding z-scores: Using a z-table or calculator, convert each percentile to its z-score.
Step 3: Plot and assess linearity: Plot the points (−1.28, 62), (−0.52, 71), (0, 76), (0.52, 82), (1.28, 94). Check whether the points fall approximately along a straight line. Here they do — no strong curvature is visible.
Answer: The points are roughly linear, so the data appear approximately normally distributed.
Why It Matters
Many AP Statistics inference procedures — such as t-tests and confidence intervals for means — require that the population be approximately normal, especially with small samples. A normal probability plot is the standard graphical tool for checking this condition on the AP exam.
Common Mistakes
Mistake: Concluding that any slight deviation from a perfect line means the data are not normal.
Correction: Small deviations and minor wobbles are expected with real data. Look for systematic patterns like strong curvature or S-shapes, not minor scatter around the line.